Large independent sets in regular graphs of large girth

Let G be a d-regular graph with girth g, and let α be the independence number of G. We show that α ( G ) ⩾ 1 2 ( 1 − ( d − 1 ) − 2 / ( d − 2 ) − ϵ ( g ) ) n where ϵ ( g ) → 0 as g → ∞ , and we compute explicit bounds on ϵ ( g ) for small g. For large g this improves previous results for all d ⩾ 7 . The method is by analysis of a simple greedy algorithm which was motivated by the differential equation method used to bound independent set sizes in random regular graphs. We use a “nibble” type of approach but require none of the sophistication of the usual nibble method arguments, relying only upon a difference equation for the expected values of certain random variables. The difference equation is approximated by a differential equation.